Problem: Divide the following complex numbers. $ \dfrac{26+7i}{-2-5i}$
Explanation: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-2+5i}$ $ \dfrac{26+7i}{-2-5i} = \dfrac{26+7i}{-2-5i} \cdot \dfrac{{-2+5i}}{{-2+5i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(26+7i) \cdot (-2+5i)} {(-2-5i) \cdot (-2+5i)} = \dfrac{(26+7i) \cdot (-2+5i)} {(-2)^2 - (-5i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(26+7i) \cdot (-2+5i)} {(-2)^2 - (-5i)^2} = $ $ \dfrac{(26+7i) \cdot (-2+5i)} {4 + 25} = $ $ \dfrac{(26+7i) \cdot (-2+5i)} {29} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({26+7i}) \cdot ({-2+5i})} {29} = $ $ \dfrac{{26} \cdot {(-2)} + {7} \cdot {(-2) i} + {26} \cdot {5 i} + {7} \cdot {5 i^2}} {29} $ Evaluate each product of two numbers. $ \dfrac{-52 - 14i + 130i + 35 i^2} {29} $ Finally, simplify the fraction. $ \dfrac{-52 - 14i + 130i - 35} {29} = \dfrac{-87 + 116i} {29} = -3+4i $